C₂×C₉×C₂₉⋊D₂₉ #44475608

order	30276
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₂₉

Aliases

direct product	C₂×C₉×C₂₉⋊D₂₉
unambiguous semidirect product	(C₂×C₂₉)⋊(C₉×D₂₉)

Splittings

Fitting	C₂×C₉×C₂₉² ⋊ C₂
derived, Sylow	C₂₉² ⋊ C₂²×C₉
Hall	C₉×C₂₉² ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₉	C₂₉⋊D₂₉
commutator, hypocenter	C₂₉²	C₂²×C₉
Frattini	C₃	C₂×C₃×C₂₉⋊D₂₉
Fitting	C₂×C₉×C₂₉²	C₂
socle	C₂×C₃×C₂₉²	C₂×C₃
prime radical	C₂×C₃×C₂₉⋊D₂₉	C₃
cosocle	C₃×C₂₉²	C₂²×C₃
LCS	G ▷ C₂₉²
UCS	C₁ ◁ C₂×C₉

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 15138

A minimal faithful permutation representation — 3 generators on 69 points:

(3,7,11,5,10,6,8,4,9)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69);(1,2)(41,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43);(12,13,15,17,19,21,23,25,27,29,31,33,35,37,39,40,38,36,34,32,30,28,26,24,22,20,18,16,14)

Conjugacy classes & character table

7596 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30276, 71);

C2×C9×C29⋊D29 #44475608